论文标题
Banach空间中某些紧凑的半流量的Poincaré-Bendixson理论
The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces
论文作者
论文摘要
我们研究半流程，以满足某些Banach空间中二次功能的某种挤压条件。在我们先前结果的某些紧凑性假设下，存在一个不变的歧管，在更限制的条件下是惯性的多种流形。在二维流形的情况下，我们获得了$ω$ limit套件的三分法的庞加莱 - 弯曲定理的类似物。此外，我们获得了存在轨道稳定的周期性轨道的条件。我们的方法统一了R.〜A.〜Smith的一系列论文，建立了与惯性歧管理论的联系，并打开了对应用的新观点。为了验证应用程序中的挤压属性，我们使用了最近开发的频率定理版本，该版本可以保证如果满足某些频域条件，则必须存在所需的二次功能。我们在$ \ mathbb {r}^{n} $和半连续性抛物线方程中介绍了结果的非线性延迟方程的应用程序，并讨论了应用程序的视角，这些视角与延迟和边界控件有关。
We study semiflows satisfying a certain squeezing condition with respect to a quadratic functional in some Banach space. Under certain compactness assumptions from our previous results it follows that there exists an invariant manifold, which is under more restrictive conditions is an inertial manifold. In the case of a two-dimensional manifold we obtain an analog of the Poincaré-Bendixson theorem on the trichotomy of $ω$-limit sets. Moreover, we obtain conditions for the existence of an orbitally stable periodic orbit. Our approach unifies a series of papers by R.~A.~Smith, establishes their connection with the theory of inertial manifolds and opens a new perspective of applications. To verify the squeezing property in applications we use recently developed versions of the frequency theorem, which guarantee the existence of the required quadratic functional if some frequency-domain condition is satisfied. We present applications of our results for nonlinear delay equations in $\mathbb{R}^{n}$ and semilinear parabolic equations and discuss perspectives of applications to parabolic problems with delay and boundary controls.
